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Unsteady MHD free convection ﬂow of Casson ﬂuid past over an oscillating vertical plate embedded in a porous medium Asma Khalid a, c, Ilyas Khan b, Arshad Khan c, Sharidan Shaﬁe c, * a

Department of Mathematics, SBK Women's University, Quetta 87300, Pakistan College of Engineering Majmaah University, Majmaah 11952, Saudi Arabia c Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Skudai 81310 UTM, Malaysia b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 October 2014 Received in revised form 10 November 2014 Accepted 2 December 2014 Available online xxx

This article studies the unsteady MHD free ﬂow of a Casson ﬂuid past an oscillating vertical plate with constant wall temperature. The ﬂuid is electrically conducting and passing through a porous medium. This phenomenon is modelled in the form of partial differential equations with initial and boundary conditions. Some suitable non-dimensional variables are introduced. The corresponding nondimensional equations with conditions are solved using the Laplace transform technique. Exact solutions for velocity and energy are obtained. They are expressed in simple forms in terms of exponential and complementary error functions of Gauss. It is found that they satisfy governing equations and corresponding conditions, and are reduced to similar solutions for Newtonian ﬂuids as a special case. Expressions for skin-friction and Nusselt number are also evaluated. Computations are carried out and the results are analysed for emerging ﬂow parameters. Copyright © 2015, Karabuk University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Casson ﬂuid MHD ﬂow Porous medium Free convection Exact solutions

1. Introduction The study of magnetohydrodynamic (MHD) ﬂow of nonNewtonian ﬂuid in a porous medium has attracted the attentions of many researchers. Of course, it is due to the fact that such phenomenons are mostly found in the optimization of solidiﬁcation processes of metals and metal alloys, the geothermal sources investigation and nuclear fuel debris treatment. However, nonNewtonian ﬂuids are subtle compare to Newtonian ﬂuids. Indeed, the resulting equations of non-Newtonian ﬂuids give highly nonlinear differential equations which are usually difﬁcult to solve. These equations add further complexities when MHD ﬂows in a porous space have been taken into account. Ample applications for the MHD ﬂows of non-Newtonian ﬂuids in a porous medium are encountered in irrigation problems, heat-storage beds, biological systems, process of petroleum, textile, paper and polymer composite industries. Numerous studies have been presented on various aspects of MHD ﬂows of non-Newtonian ﬂuid ﬂows passing through a porous medium. One may refer to some recent investigations [1e5]. * Corresponding author. Tel.: þ60 137731773. E-mail address: [email protected] (S. Shaﬁe). Peer review under responsibility of Karabuk University.

On the other hand, convection ﬂow arises in many physical situations such as in the cooling of nuclear reactors and in the study of environmental heat transfer processes amongst others. Convection is of three types namely free, mixed and force. Amongst them free convection is important in many engineering applications including an example of automatic control systems consist of electrical and electronic components, regularly subjected to periodic heating and cooled by free convection process. Some recent studies containing the free convection phenomenon can be found in [6e10] and the references therein. Besides that the work on free convection for non-Newtonian ﬂuids when exact solutions are needed is limited. Further, when MHD and porosity effects are added to the governing equations then even such solutions are scarce. Farhad et al. [11] obtained closed form solutions for unsteady free convection ﬂow of a second grade ﬂuid over an oscillating vertical plate. Khan et al. [12] developed exact solutions for unsteady free convection ﬂow of Walters'-B ﬂuid. Samiulhaq et al. [13] analysed unsteady MHD free convection ﬂow of a second grade ﬂuid in a porous medium with ramped wall temperature. In nature, some non-Newtonian ﬂuids behave like elastic solid that is, no ﬂow occur with small shear stress. Casson ﬂuid is one of such ﬂuids. This ﬂuid has distinct features and is quite famous recently. Casson ﬂuid model was introduced by Casson in 1959 for the prediction of the ﬂow behaviour of pigment-oil suspensions

http://dx.doi.org/10.1016/j.jestch.2014.12.006 2215-0986/Copyright © 2015, Karabuk University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: A. Khalid, et al., Unsteady MHD free convection ﬂow of Casson ﬂuid past over an oscillating vertical plate embedded in a porous medium, Engineering Science and Technology, an International Journal (2015), http://dx.doi.org/10.1016/ j.jestch.2014.12.006

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A. Khalid et al. / Engineering Science and Technology, an International Journal xxx (2015) 1e9

[14]. So, for the ﬂow, the shear stress magnitude of Casson ﬂuid needs to exceed the yield shear stress, otherwise the ﬂuid behaves as a rigid body. This type of ﬂuids can be marked as a purely viscous ﬂuid with high viscosity [15]. Casson model is based on a structure model of the interactive behavior of solid and liquid phases of a twophase suspension. Some famous examples of Casson ﬂuid include jelly, tomato sauce, honey, soup and concentrated fruit juices. Human blood can also be treated as Casson ﬂuid due to the presence of several substances such as protein, ﬁbrinogen, globulin in aqueous base plasma and human red blood cells [16,17]. In the earlier studies on Casson ﬂuid, Boyd et al. [18] discussed the steady and oscillatory ﬂow of blood by taking into account Casson ﬂuid whereas Fredrickson [19] investigated its steady ﬂow in a tube. The peristaltic ﬂow of a Casson ﬂuid in a two-dimensional channel is described by Mernone et al. [20]. Mustafa et al. [21] studied the unsteady boundary layer ﬂow and heat transfer of a Casson ﬂuid over a moving ﬂat plate with a parallel free stream using homotopy analysis method (HAM). Mixed convection stagnation-point ﬂow of Casson ﬂuid with convective boundary conditions is examined by Hayat et al. [22]. Shaw et al. [23] discussed the effect of non-Newtonian characteristics of blood on magnetic targeting in the impermeable micro-vessel. Magnetic targeting in the impermeable microvessel with two-phase ﬂuid model-NonNewtonian characteristic of blood carried out by Shaw and Murthy [24]. Pulsatile Casson ﬂuid ﬂow through a stenosed bifurcated artery also studied by Shaw et al. [25]. The effects of thermal radiation on Casson ﬂuid ﬂow and heat transfer over an unsteady stretching surface subjected to suction/blowing has been developed by Mukhopadhyay [26]. Bhattacharyya [27] constructed the boundary layer stagnation-point ﬂow of Casson ﬂuid and heat transfer towards a shrinking/stretching sheet. Mukhopadhyay et al. [28] also analysed the Casson ﬂuid ﬂow over an unsteady stretching surface followed by Pramanik [29] where he studied the Casson ﬂuid ﬂow and heat transfer past an exponentially porous stretching surface in the presence of thermal radiation. In all of the above studies the solutions of Casson ﬂuid are either obtained by using approximate method or by any numerical scheme. There are very few cases in which the exact analytical solutions of Casson ﬂuid are obtained. These solutions are even rare when Casson ﬂuid in free convection ﬂow with constant wall temperature is considered. On the other hand, the ﬂow of Casson ﬂuids (such as drilling muds, clay coatings and other suspensions, certain oils and greases, polymer melts, blood and many emulsions), in the presence of heat transfer is an important research area due to its relevance in the optimized processing of chocolate, toffee, and other foodstuffs [21,30e32]. The purpose of the present investigation is two-fold. Firstly, it incorporates the effects of magnetic ﬁeld by considering the ﬂuid to be electrically conducting. Secondly, the ﬂuid is considered in a porous medium. More exactly, the present work concentres on unsteady MHD free convection ﬂow of a Casson ﬂuid over a vertical plate embedded in a porous medium. Exact solutions when the plate performs sine and cosine oscillations with constant wall temperature are obtained by using the Laplace transform technique [33e36]. Analytical and numerical results for skin-friction and Nusselt number are provided. Graphical results are presented and discussed for various physical parameters entering into the problem. 2. Formulation of the problem We consider Casson ﬂuid over an inﬁnite vertical ﬂat plate embedded in a saturated porous medium. The ﬂow being conﬁned to y > 0, where y is the coordinate measured in the normal direction to the plate. The ﬂuid is assumed to be electrically conducting with a uniform magnetic ﬁeld B of strength B0 , applied in a direction

perpendicular to the plate. The magnetic Reynolds number is assumed to be small enough to neglect the effects of applied magnetic ﬁeld. Initially, for time t ¼ 0; both the ﬂuid and the plate are at rest with uniform temperature. At time t ¼ 0þ the plate begins to oscillate in its plane ðy ¼ 0Þ according to

V ¼ UHðtÞcosðutÞi; or V ¼ U sinðutÞi; t > 0;

(1)

where the constant U is the amplitude of the plate oscillations, HðtÞ is the unit step function, i is the unit vector in the vertical ﬂow direction and u is the frequency of oscillation of the plate. At the same time, the plate temperature is raised to Tw which is thereafter maintained constant. The rheological equation of state for the Cauchy stress tensor of Casson ﬂuid is written as, (see [22,26,27,28,32])

t ¼ t0 þ mg ; or

8 py > > p ﬃﬃﬃﬃﬃﬃ eij ; p > pc þ 2 m > B < 2p tij ¼ ; > py > > : 2 mB þ pﬃﬃﬃﬃﬃﬃﬃﬃ eij ; p < pc 2pc where p ¼ eij eij and eij is the ði; jÞih component of the deformation rate, p is the product of the component of deformation rate with itself, pc is a critical value of this product based on the nonNewtonian model, mB is plastic dynamic viscosity of the nonNewtonian ﬂuid and py is yield stress of ﬂuid. Before we derive the governing equations, the following assumptions are made, rigid plate, incompressible ﬂow, unsteady ﬂow, unidirectional ﬂow, one dimensional ﬂow, non-Newtonian ﬂow, free convection, oscillating vertical plate and viscous dissipation term in the energy equation is neglected. Under these conditions we get the following set of partial differential equations

r

vu 1 v2 u m4 ¼ mB 1 þ sB20 u u þ rgbðT T∞ Þ; vt g vy2 k1

rcp

vT v2 T ¼k 2; vt vy

(2) (3)

together with initial and boundary conditions

t < 0 : u ¼ 0; T ¼ T∞ for all y > 0; t 0 : u ¼ UHðtÞcosðutÞ or u ¼ U sinðutÞ; T ¼ Tw at y ¼ 0; u/0; T/T∞ as y/∞; (4)

where u; t; T; mB ; g; r; g; b; cp ; k; s; f; and k1 are the velocity of the ﬂuid in x direction, time, temperature, plastic dynamic viscosity, Casson parameter, the constant density, the gravitational acceleration, volumetric coefﬁcient of thermal expansion, speciﬁc heat at constant pressure and thermal conductivity, electric conductivity of the ﬂuid, porosity and permeability of the ﬂuid, respectively. We introduce the following dimensionless variables

u* ¼

u * U U2 T T∞ un t ; y ¼ y; t * ¼ t; q ¼ ; u* ¼ 2 ; t* ¼ 2 ; U n n Tw T∞ U ru (5)

into Equations. (2)e(4), and we get (* symbols are dropped for simplicity)

vu ¼ vt Pr

1þ

1 v2 u 1 M 2 u* u* þ Grq; g vy2 K

vq v2 q ¼ ; vt vy2

(6) (7)

with associated initial and boundary conditions

Please cite this article in press as: A. Khalid, et al., Unsteady MHD free convection ﬂow of Casson ﬂuid past over an oscillating vertical plate embedded in a porous medium, Engineering Science and Technology, an International Journal (2015), http://dx.doi.org/10.1016/ j.jestch.2014.12.006

A. Khalid et al. / Engineering Science and Technology, an International Journal xxx (2015) 1e9

t < 0 : u ¼ 0; q ¼ 0 for all y > 0; t 0 : u ¼ HðtÞcosðutÞ or u ¼ sinðutÞ; q ¼ 1 at y ¼ 0; u/0; q/0 as y/∞;

(8)

qðy; tÞ ¼ erfc

y 2

rﬃﬃﬃﬃﬃ! Pr ; t

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y a aðLiuÞ y aðLiuÞ e ðL iuÞt þ e þ ðL iuÞt erfc erfc uc ðy; tÞ ¼ e 4 2 t 2 t ! !# " r ﬃﬃﬃ r ﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y a aðLþiuÞ y aðLþiuÞ þ e ðL þ iuÞt þ e þ ðL þ iuÞt erfc erfc e 4 2 t 2 t " ! ! ! ! !# rﬃﬃﬃ rﬃﬃﬃ rﬃﬃﬃ rﬃﬃﬃ! pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ ab y a ypﬃﬃﬃﬃ y a y a ypﬃﬃﬃﬃ y a aL aL t e Lt þ tþ e þ Lt þ erfc erfc 2 2 L 2 t 2 L 2 t ! " # rﬃﬃﬃﬃﬃ rﬃﬃﬃ pﬃﬃﬃﬃﬃ t Pry2 Pry2 y Pr y Pr ab t þ erfc e 4t : 2 t p 2

3

(11)

(12)

The subscript “c” on the left side of Equation (12) stands for the cosine oscillations of the plate. Similarly, the velocity corresponding to the sine oscillations of the plate is given by

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 y a y a ðL iuÞt þ ey aðLiuÞ erfc þ ðL iuÞt us ðy; tÞ ¼ eiut ey aðLiuÞ erfc 4i 2 t 2 t " rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y a aðLþiuÞ y aðLþiuÞ ðL þ iuÞt þ e þ ðL þ iuÞt þ e erfc erfc e 4i 2 t 2 t " ! !# rﬃﬃﬃ! rﬃﬃﬃ! rﬃﬃﬃ! rﬃﬃﬃ! pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ ab y a ypﬃﬃﬃﬃ y a y a ypﬃﬃﬃﬃ y a aL aL e Lt þ e þ Lt þ erfc erfc t tþ 2 2 L 2 t 2 L 2 t ! # " r ﬃﬃﬃﬃﬃ r ﬃﬃﬃ pﬃﬃﬃﬃﬃ t Pry2 Pry2 y Pr y Pr erfc e 4t ; ab t þ 2 t p 2

(13)

where

where

mcp 2 syB20 1 y42 ygbðTw T∞ Þ ;M ¼ ; ¼ ; Gr ¼ and g 2 k U3 rU K k1 U 2 pﬃﬃﬃﬃﬃﬃﬃﬃ m 2pc ; ¼ B Py

g Gr ;b ¼ : 1þg Pr 1

Pr ¼

a¼

here Pr is the Prandtl number, M is the magnetic parameter called Hartmann number, K is the dimensionless permeability parameter, Gr is the Grashof number and g is the Casson parameter.

Note that the above solutions for velocity are only valid for Prs1. Moreover, the solution for Pr ¼ 1 can be easily obtained by putting Pr ¼ 1 into Equation (7), and follow a similar procedure as discussed above. The obtained solutions for cosine and sine oscillations of the plate when Pr ¼ 1 are

3. Exact solutions

uc ðy; tÞ ¼

In order to ﬁnd exact solutions of the system of Equations. (6)e(8), we use the Laplace transform technique. Thus by taking the Laplace transforms of Equations. (6) and (7), using initial and boundary conditions (8), we get the following solutions in the transformed ðy; qÞ plane

pﬃﬃﬃﬃﬃﬃ 1 qðy; qÞ ¼ ey Prq ; q pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ab pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ab q uðy; qÞ ¼ 2 ey aðqþLÞ þ 2 ey aðqþLÞ 2 ey Prq : 2 q þu q q

(9) (10)

The inverse Laplace transforms of Equations. (9) and (10) are obtained as follows:

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a aðLiuÞ e ðL iuÞt erfc e 4 2 t rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y aðLiuÞ þe þ ðL iuÞt erfc 2 t " rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a aðLþiuÞ þ e ðL þ iuÞt erfc e 4 2 t rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a þ ey aðLþiuÞ erfc þ ðL þ iuÞt 2 t " rﬃﬃﬃ # a Gr y t y2 y 4t 2 e y erf c pﬃﬃ ; 2 p 2 t

(14)

Please cite this article in press as: A. Khalid, et al., Unsteady MHD free convection ﬂow of Casson ﬂuid past over an oscillating vertical plate embedded in a porous medium, Engineering Science and Technology, an International Journal (2015), http://dx.doi.org/10.1016/ j.jestch.2014.12.006

4

A. Khalid et al. / Engineering Science and Technology, an International Journal xxx (2015) 1e9

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 y a y a ðL iuÞt þ ey aðLiuÞ erfc þ ðL iuÞt us ðy; tÞ ¼ eiut ey aðLiuÞ erfc 4i 2 t 2 t " rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 y a y a ðL þ iuÞt þ ey aðLþiuÞ erfc þ ðL þ iuÞt þ eiut ey aðLþiuÞ erfc 4i 2 t 2 t " rﬃﬃﬃ # a Gr y t y2 y 4t 2 y erf c pﬃﬃ : e 2 p 2 t

Note that in Equations. (12) and (13), the ﬁrst two terms in each equation account the contribution from mechanical parts while the last two terms show the thermal effects. On the other hand in Equations. (14) and (15), the last term in each equation shows the contribution from the thermal part.

4.1. Solutions for large values of g By taking g/∞ into Equations. (12) and (13), the corresponding solutions for viscous ﬂuid can be obtained as a special case:

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y 1 y 1 ðLiuÞ y ðLiuÞ e ðL iuÞt þ e þ ðL iuÞt erfc erfc uc ðy; tÞ ¼ e 4 2 t 2 t ! " rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y 1 y 1 ðLþiuÞ y ðLþiuÞ e ðL þ iuÞt þ e þ ðL þ iuÞt þ erfc erfc e 4 2 t 2 t " ! !# rﬃﬃﬃ! rﬃﬃﬃ! rﬃﬃﬃ! rﬃﬃﬃ! pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ b y 1 ypﬃﬃL y 1 y 1 ypﬃﬃL y 1 þ t erfc erfc e Lt þ e þ Lt tþ 2 2 L 2 t 2 L 2 t # " rﬃﬃﬃﬃﬃ! rﬃﬃﬃ 2 2 p ﬃﬃﬃﬃﬃ Pry y Pr t Pry y Pr erfc e 4t ; b tþ 2 t p 2

3.1. Nusselt number and skin-friction Expressions for Nusselt number and skin-friction are calculated from Equations. (11) and (12) using the relations

rﬃﬃﬃﬃﬃ n vT * Pr ; Nu ¼ ¼ * * U pt ðT T Þ vy ∞ y ¼0 0 1 vu t ¼ m 1 þ ; # " g vyy¼0 qﬃﬃﬃﬃ qﬃﬃ qﬃﬃ pﬃﬃﬃﬃﬃpﬃﬃ Lt a Lt a Pr 2 e H t þ ab Pr t abe t t þ ab t tÞ 1 pﬃﬃﬃ t¼ ½ 2 p rﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ a þ 2 aLt erfð Lt Þ ab½ L qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eiut Hf aðL iuÞ erfð tðL iuÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ e2itu aðL þ iuÞ erfð tðL þ iuÞg:

(16)

(17)

(15)

(18)

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y 1 ðLiuÞ ðL iuÞt erfc us ðy; tÞ ¼ e e 4i 2 t rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y 1 þ ðL iuÞt þ ey ðLiuÞ erfc 2 t " rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 y 1 ðL þ iuÞt þ eiut ey ðLþiuÞ erfc 4i 2 t rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y 1 y ðLþiuÞ þ ðL þ iuÞt þe erfc 2 t " ! rﬃﬃﬃ! rﬃﬃﬃ! pﬃﬃﬃﬃﬃ b y 1 ypﬃﬃL y 1 þ erfc t e Lt 2 2 L 2 t !# rﬃﬃﬃ! rﬃﬃﬃ! pﬃﬃﬃﬃﬃ y 1 ypﬃﬃL y 1 erfc e þ Lt þ tþ 2 L 2 t ! # " rﬃﬃﬃﬃﬃ rﬃﬃﬃ pﬃﬃﬃﬃﬃ t Pry2 Pry2 y Pr 4t y Pr erfc : e b tþ 2 t p 2 (19)

4. Special cases In order to underline the theoretical value of the general solutions (12) and (13) for velocity, as well as to gain physical insight of the ﬂow regime, we consider some special cases whose technical relevance is well known in the literature.

4.2. Solutions for stokes ﬁrst problem By taking u ¼ 0, which corresponds to impulsive motion of the plate, then Equations. (12) and (13), yield

Please cite this article in press as: A. Khalid, et al., Unsteady MHD free convection ﬂow of Casson ﬂuid past over an oscillating vertical plate embedded in a porous medium, Engineering Science and Technology, an International Journal (2015), http://dx.doi.org/10.1016/ j.jestch.2014.12.006

A. Khalid et al. / Engineering Science and Technology, an International Journal xxx (2015) 1e9

" ! !# rﬃﬃﬃ rﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ 1 y a pﬃﬃﬃﬃﬃ y a pﬃﬃﬃﬃﬃ Lt þ ey aL erfc þ Lt uc ðy;tÞ ¼ H ey aL erfc 2 2 t 2 t " ! rﬃﬃﬃ! rﬃﬃﬃ! pﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ ab y a y aL y a erfc t e Lt þ 2 2 L 2 t !# rﬃﬃﬃ! rﬃﬃﬃ! pﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ y a y aL y a e þ Lt erfc þ tþ 2 L 2 t 2 rﬃﬃﬃﬃﬃ! pﬃﬃﬃﬃﬃpﬃﬃ Pry2 3 2 Pry y Pr y Pr t e 4t 5 pﬃﬃﬃ ab4 t þ ; erfc 2 t 2 p (20) " ! !# r ﬃﬃﬃ r ﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ 1 y a pﬃﬃﬃﬃﬃ y a pﬃﬃﬃﬃﬃ us ðy;tÞ ¼ H ey aL erfc Lt þey aL erfc þ Lt 2i 2 t 2 t " ! rﬃﬃﬃ! rﬃﬃﬃ! pﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ ab y a y aL y a t e Lt erfc þ 2 2 L 2 t !# rﬃﬃﬃ! pﬃﬃﬃﬃ rﬃﬃﬃ! pﬃﬃﬃﬃﬃ y a y aL y a e þ Lt erfc þ tþ 2 L 2 t 2 3 rﬃﬃﬃﬃﬃ! 2 pﬃﬃ 2 pﬃﬃﬃﬃﬃ t ePry4t Pry y Pr ab4 t þ y Pr pﬃﬃﬃ 5: erfc 2 t 2 p

5

4.4. Solution in the absence of mechanical effects Let us now assume that the inﬁnite plate is kept at rest all the time. In this case, the wall velocity of the ﬂuid is zero for each real value of t and thus the mechanical component of velocity identically vanishes. Consequently, the velocity of the ﬂuid uðy; tÞ reduces to the thermal component of Equation (13). Its temperature as well as the surface heat transfer rate are given by the same equalities (11) and (16).

4.5. Solution in the absence of mechanical effects

respectively,

(21)

4.3. Absence of MHD and porosity effects: attend The temperature distribution is not effected by MHD and porous medium, as it results from Equation (11). However, MHD and porosity have strong inﬂuence on velocity as it can be seen from the mechanical parts of Equations. (12) and (13). Thus in the absence of MHD ðM ¼ 0Þ and porous medium ð1=K ¼ 0Þ, these equalities become

In the last case, we assume that the ﬂow is induced only due to bounding plate and the corresponding buoyancy forces are zero equivalently it shows the absence of free convection ðGr ¼ 0Þ due to the differences in temperature gradient. This shows that the thermal parts of velocities in Equations (12) and (13) are zero. Hence the ﬂow is only governed by the corresponding mechanical parts given by

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a aðLiuÞ e ðL iuÞt erfc uc ðy; tÞ ¼ e 4 2 t rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y aðLiuÞ þ ðL iuÞt þe erfc 2 t ! " r ﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a aðLþiuÞ e ðL þ iuÞt þ erfc e 4 2 t rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y aðLþiuÞ þ ðL þ iuÞt ; þe erfc 2 t

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y a aðiuÞ y aðiuÞ e ð iuÞt þ e þ ð iuÞt erfc erfc uc ðy; tÞ ¼ e 4 2 t 2 t ! " rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ HðtÞ iut ypﬃﬃﬃﬃﬃﬃﬃﬃ y a y a aðiuÞ y aðiuÞ e ðiuÞt þ e þ ðiuÞt þ erfc erfc e 4 2 t 2 t " ! ! !# rﬃﬃﬃ rﬃﬃﬃ! pﬃﬃ pﬃﬃ ab y pﬃﬃﬃ ypﬃﬃa y a y pﬃﬃﬃ ypﬃﬃa y a t a e t þ tþ a e þ t þ erfc erfc 2 2 2 t 2 2 t " # rﬃﬃﬃﬃﬃ! rﬃﬃﬃ pﬃﬃﬃﬃﬃ t Pry2 Pry2 y Pr y Pr ab t þ erfc e 4t ; 2 t p 2

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y a aðiuÞ y aðiuÞ ð iuÞt þ e þ ð iuÞt erfc erfc us ðy; tÞ ¼ e e 4i 2 t 2 t ! " rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 iut ypﬃﬃﬃﬃﬃﬃﬃﬃ y a y a aðiuÞ y aðiuÞ ðiuÞt þ e þ ðiuÞt þ e erfc erfc e 4i 2 t 2 t " ! ! !# rﬃﬃﬃ rﬃﬃﬃ! pﬃﬃ pﬃﬃ ab y pﬃﬃﬃﬃﬃ ypﬃﬃa y a y pﬃﬃﬃ ypﬃﬃa y a tþ t a1 e t þ a e þ t þ erfc erfc 2 2 2 t 2 2 t " ! # rﬃﬃﬃﬃﬃ rﬃﬃﬃ pﬃﬃﬃﬃﬃ t Pry2 Pry2 y Pr ab t þ y Pr erfc e 4t : 2 t p 2

(24)

(22)

(23)

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A. Khalid et al. / Engineering Science and Technology, an International Journal xxx (2015) 1e9

" rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a aðLiuÞ ðL iuÞt erfc us ðy; tÞ ¼ e e 4i 2 t rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y aðLiuÞ þe erfc þ ðL iuÞt 2 t " rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 iut ypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a aðLþiuÞ ðL þ iuÞt erfc e þ e 4i 2 t rﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!# pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ y a y aðLþiuÞ þ ðL þ iuÞt : þe erfc 2 t (25) Note that Equations (24) and (25) when g/∞; are identical to those obtained by Fetecau et al. [31], see Equations (8) and (9). This fact is also shown in Fig. 10.

5. Results and discussion In this section, the obtained exact solutions are studied numerically in order to determine the effects of several involved parameters such as Prandtl number Pr, Grashof number Gr, Casson parameter g, magnetic parameter M, permeability of porous medium K, phase angle ut and time t. Numerical values of skin-friction and Nusselt number are computed and presented in tables for different parameters. Physical sketch of the problem is shown in Fig. 1. Fig. 2 exhibits the velocity proﬁles for different values of Prandtl number Pr, when the other parameters are ﬁxed. It is observed that velocity of the ﬂuid decreases with increasing Prandtl number. Fig. 3 illustrates the proﬁles of velocity for different values of Gr: It is observed that velocity increases with increasing values of Gr: The ﬂow is accelerated due to the enhancement in the buoyancy forces corresponding to the increasing values of Grashof number, i.e., free convection effects. The inﬂuence of Casson ﬂuid parameter on velocity proﬁles is shown in Fig. 4. It is found that velocity decreases with increasing values of g. It is important to note that an increase in Casson parameter makes the velocity boundary layer thickness shorter. It is further observed from this graph that when the Casson paramter g is large enough i.e. g/∞, the non-Newtonian behaviours disappear and the ﬂuid purely

Fig. 1. Physical sketch of the problem.

Fig. 2. Proﬁles of velocity for different M ¼ 0:5; K ¼ 0:2; u ¼ p=4; t ¼ 0:2 and Gr ¼ 3:

values

of

Pr,

when

Fig. 3. Proﬁles of velocity for different Pr ¼ 0:3; g ¼ 0:6; M ¼ 0:5; K ¼ 0:2; t ¼ 0:3 and u ¼ p=4:

values

of

Gr,

when

behaves like a Newtonian ﬂuid. Thus, the velocity boundary layer thickness for Casson ﬂuid is larger than the Newtonian ﬂuid. It occurs because of plasticity of Casson ﬂuid. When Casson parameter decreases the plasticity of the ﬂuid increases, which causes the

Fig. 4. Proﬁles of velocity for different Pr ¼ 0:3;Gr ¼ 0; M ¼ 0:2; K ¼ 2; t ¼ 0:3 and u ¼ p=4:

values

of

g,

when

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A. Khalid et al. / Engineering Science and Technology, an International Journal xxx (2015) 1e9

Fig. 5. Proﬁles of velocity for different Gr ¼ 3; Pr ¼ 0:3; M ¼ 0:5; K ¼ 0:2; t ¼ 1 and g ¼ 0:5:

values

of

ut,

when

Fig. 6. Proﬁles of velocity for different Pr ¼ 0:3;Gr ¼ 3; g ¼ 0:5; K ¼ 0:2; t ¼ 0:3 and u ¼ p=4:

values

of

M,

when

increment in velocity boundary layer thickness. The graphical results for the phase angle ut; are shown in Fig. 5. It is observed that the ﬂuid is oscillating between 1 and 1. These ﬂuctuations near the plate are maximum and decrease for further values of

Fig. 7. Proﬁles of velocity for different Gr ¼ 3; g ¼ 0:5; M ¼ 0:5; t ¼ 0:3 and u ¼ p=4:

values

of

K,

when

Fig. 8. Proﬁles of velocity for different Pr ¼ 0:3;Gr ¼ 0; g ¼ 0:5; M ¼ 0:5; K ¼ 1 and u ¼ 0:

7

values

of

t,

when

Fig. 9. Proﬁles of temperature for different values of Pr, when t ¼ 0.4.

independent variable y. This ﬁgure can easily help us to check the accuracy of our results. For illustration of such results we have concentrated more on the values of ut ¼ 0; p=2 and p. We can see that for these values of ut; the velocity shows its value either 1,

Pr ¼ 0:3; Fig. 10. Proﬁles of temperature for different values of t, when Pr ¼ 0.71.

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A. Khalid et al. / Engineering Science and Technology, an International Journal xxx (2015) 1e9

Table 1 Skin-friction variations. Pr

Gr

g

ut

M

K

t

t

0.3 0.71 0.3 0.3 0.3 0.3 0.3 0.3

3 3 5 3 3 3 3 3

0.5 0.5 0.5 1.0 0.5 0.5 0.5 0.5

p/4 p/4 p/4 p/4 p/2 p/4 p/4 p/4

0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5

0.2 0.2 0.2 0.2 0.2 0.2 1.0 0.2

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.5

1.02992 1.29166 0.856995 0.93952 1.05009 1.08472 0.670207 0.692367

0 or 1 which are identical with the imposed boundary conditions of velocity in Equation (9). Hence, both the graphical and mathematical results are found in excellent agreement. Fig. 6 displays the effect of magnetic parameter M on the velocity proﬁles. It is observed that the amplitude of the velocity as well as the boundary layer thickness decreases when M is increased. Physically, it may also be expected due to the fact that the application of a transverse magnetic ﬁeld results in a resistive type force (called Lorentz force) similar to the drag force, and upon increasing the values of M, the drag force increases which leads to the deceleration of the ﬂow. In Fig. 7, the proﬁles of velocity have been plotted for various values of permeability parameter K by keeping other parameters ﬁxed. It is observed that for large values of K, velocity and boundary layer thickness increase which explains the physical situation that as K increases, the resistance of the porous medium is lowered which increases the momentum development of the ﬂow regime, ultimately enhances the velocity ﬁeld. In Fig. 8 the inﬂuence of dimensionless time t on the velocity proﬁles is shown. It is found that the velocity is an increasing function of time t. It is depicted from Fig. 9 that, the temperature decreases as the Prandtl number Pr increases. It is justiﬁed due to the fact that thermal conductivity of the ﬂuid decreases with increasing Prandtl number Pr and hence decreases the thermal boundary layer thickness. Fig. 10 is plotted to show the effects of the dimensionless time t on the temperature proﬁles. Four different values of time t ¼ 0.1, t ¼ 0.2, t ¼ 0.3 and t ¼ 0.1 are chosen. Obviously the temperature increases with increasing time t. This graphical behaviour of temperature is in good agreement with the corresponding boundary conditions of temperature proﬁles as shown in Equation (8). Results for skin-friction and Nusselt number are computed in Tables 1 and 2. The computations of skin-friction give complex results. Therefore, for the sake of convenience we have considered in Table 1 only its real part. The inﬂuence of Casson parameter on velocity and skin-friction is found identical with the published results of Mukhopadhyay [[23], see Figs. 3(a) and 7(a)]. Table 1 shows that skin-friction increases with increasing values of Pr and ut whereas it decreases with increasing values of Gr, g and t. On the other hand, it is found from Table 2 that Nusselt number increases with increasing Pr whereas decreases with increasing t: For the veriﬁcation, we have compared our results with those of Fetecau et al. [31]. This comparison is shown in Fig. 11. It is found that our limiting solution (24) when g/∞; (graph shown by solid line) are identical to Equation (8) (graph shown by ﬁlled squares) obtained by Fetecau et al. [31]. This conﬁrms the accuracy of our

Fig. 11. Comparison of the present result [see Equation (24), when g/∞] with that obtained by Fetecau et al. [31], [see Equation (8)], when t ¼ 0:2; u ¼ 0; a ¼ 1; U ¼ 1 and v ¼ 1:

obtained results. This ﬁgure further shows the comparison of velocity proﬁles in the absence as well as in the presence of MHD and porous medium. The graph for velocity when M ¼ 0:5 and K ¼ 0:2 are shown by ﬁlled circles whereas the graph for present velocity when M ¼ 1=K ¼ 0 are zero is given by solid line. It is clearly seen that the velocity decays early in the presence of MHD and porous medium. 6. Conclusion In this paper an exact analysis is performed to investigate the unsteady boundary layer ﬂow of a Casson ﬂuid past an oscillating vertical plate with constant wall temperature. The dimensionless governing equations are solved by using the Laplace transform technique. The results for velocity and temperature are obtained and plotted graphically. The numerical results for skin-friction and Nusselt number are computed in tables. The main conclusions of this study are as follows: 1. Velocity increases with increasing Gr; K and t whereas decreases with increasing values of Pr; M; g and ut. 2. Temperature increases with increasing t whereas decreases when Pr; is increased. 3. Skin-friction increases with increasing values of Pr;M and ut whereas it decreases with increasing values of Gr; g; K and t. 4. Nusselt number increases with increasing Pr whereas decreases with increasing t: 5. Solution (24) is found in excellent agreement with those obtained by Fetecau et al. [31]. Acknowledgments The authors would like to acknowledge the SBKWU (HEC) Pakistan, Ministry of Education Malaysia (MOE) and Research Management Centre-UTM for the ﬁnancial support through vote numbers 06H67 and 4F255 for this research. References

Table 2 Nusstle number variations. Pr

t

Nu

0.3 0.71 0.3

0.3 0.3 0.6

0.564 0.867 0.398

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